Algorithm 2: Construction of the Measurement-Formation Path

Input: Sparse measurement coefficients vectors ${\Phi }_1,{\Phi }_2\dots ,{\Phi }_M$ each with n i.i.d. elements; nodes’ maximum communication range Rcomm;

Output: Set of measurement formation a trees $T=\left\{T_1,T_2,\dots ,T_M\right\}$, $T_i=\left\{I{n}_i,S_i,Y^i\right\}$

1 For each measurement $i=1, 2,\dots , M$

2 Calculate the set of interested nodes $I{n}_i=\left\{all nodes s.t. {\Phi }_{ij}\ne 0, j=1, 2,\dots ,n\right\}$.

3 Calculate the set of neighbor nodes $Ne=\left\{each node v s.t. dist\left(v,t\right)\le R_{comm}, v\in V, t\in I{n}_i\right\}$.

4 Do

5 Calculate the number of edges incident from $Ne$ to $I{n}_i$ (denoted as $De\left\{\left|Ne\right|\right\}$).

6 If $Max\left(De\left\{\left|Ne\right|\right\}\right)\ge 2$

7 Remove all adjacent nodes of node v from $I{n}_i$ where $v\in I{n}_i$ and $De\left\{v\right\}=Max\left(De\left\{\left|Ne\right|\right\}\right)$

8 Add node v into Ini

9 Renew Ne and Ini

10 Ifend

11 While ($\left|I{n}_i^{new}\right|<\left|I{n}_i^{old}\right|$).

12 Calculate the convex hull (denoted as $Cl$) of set $Ne\cup \left\{sink\right\}$ by using Graham scan method

13 Construct the MST which takes the sink as a root and connects all Cl nodes

14 $T_i=\left\{I{n}_i,Cl,Y^i\right\}$ where $Y^i$ recorded the edge information of such MST

15 Forend

16 Return $T=\left\{T_1,\dots ,T_M\right\}$